Compression (functional analysis)

In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operator : P_K T \vert_K : K \rightarrow K , where P_K : H \rightarrow K is the orthogonal projection onto K. This is a natural way to obtain an operator on K from an operator on the whole Hilbert space. If K is an invariant subspace for T, then the compression of T to K is the restricted operator K→K sending k to Tk. More generally, for a linear operator T on a Hilbert space H and an isometry V on a subspace W of H , define the compression of T to W by : T_W = V^*TV : W \rightarrow W , where V^* is the adjoint of V. If T is a self-adjoint operator, then the compression T_W is also self-adjoint. When V is replaced by the inclusion map I: W \to H , V^* = I^*=P_K : H \to W , and we acquire the special definition above.

In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operator P_K T \vert_K : K \rightarrow K , where P_K : H \rightarrow K is the orthogonal projection onto K. This is a natural way to obtain an operator on K from an operator on the whole Hilbert space. If K is an invariant subspace for T, then the compression of T to K is the restricted operator K→K sending k to Tk. More generally, for a linear operator T on a Hilbert space H and an isometry V on a subspace W of H , define the compression of T to W by T_W = V^*TV : W \rightarrow W , where V^* is the adjoint of V. If T is a self-adjoint operator, then the compression T_W is also self-adjoint. When V is replaced by the inclusion map I: W \to H , V^* = I^*=P_K : H \to W , and we acquire the special definition above. ==See also== Dilation (operator theory) ==References== P. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, 1982.

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